$\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$_2$. We consider its root system as follows (though it is probably not necessary to state the question): \begin{equation*} \Delta^+(\mathfrak g,\mathfrak h) = \left\{ \begin{array}{ll} \ee_2-\ee_3,&\ee_1-2\ee_2+\ee_3,\\ \ee_1-\ee_2,&\ee_1+\ee_2-2\ee_3,\\ \ee_1-\ee_3,&2\ee_1-\ee_2-\ee_3 \end{array} \right\}, \end{equation*} where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak g$ with $\mathfrak h^*=\{\sum_{i=1}^3 a_i\ee_i: a_i\in\mathbb C,\; \sum_{i=1}^3a_i=0\}$. We will use its root decomposition \begin{equation*} \mathfrak g=\mathfrak h\oplus \bigoplus_{\alpha\in\Delta(\mathfrak g,\mathfrak h)} \mathfrak g_\alpha. \end{equation*}
Let $(\pi,V_\pi)$ be a non-trivial irreducible representation of $\mathfrak g$. Suppose $v\in V_\pi$ is in the weight space of weight zero (i.e. $v\in V_\pi(0)$) and satisfies \begin{equation*} \pi(X_{\ee_1-\ee_2})(v)=0 \quad\text{ and }\quad \pi(X_{\ee_1+\ee_2-2\ee_3})(v)=0, \end{equation*} where $X_{\ee_1-\ee_2}\in\mathfrak g_{\ee_1-\ee_2}$ and $X_{\ee_1+\ee_2-2\ee_3}\in \mathfrak g_{\ee_1+\ee_2-2\ee_3}$ are non-trivial.
Can we ensure that $v=0$?
Note that the story would be very different if we replace the roots $\ee_1-\ee_2$ and $\ee_1+\ee_2-2\ee_3$ by any pair $\alpha,\beta\in\Delta(\mathfrak g,\mathfrak h)$ with $\alpha$ short, $\beta$ long and non-orthogonal. In that case, one easily obtains that $\pi(X_{a\alpha+b\beta})(v)=0$ for all $a,b\in\mathbb Z$, which implies that $\mathbb C v$ is $\mathfrak g$-invariant since $\mathbb Z\alpha+\mathbb Z\beta=\operatorname{Span}_\mathbb Z\Delta(\mathfrak g,\mathfrak h)$, contradicting the assumptions on $\pi$.
